Quadratic Functions. and Equations

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1 Name: Quadratic Functions and Equations 1. + x 2 is a parabola 2. - x 2 is a parabola 3. A quadratic function is in the form ax 2 + bx + c, where a and is the y-intercept 4. Equation of the Axis of Symmetry is x = 5. Solution(s), Root(s), Zero(s) are the values of the points where the graph intersects the x-axis

2 Quadratic Functions and Equations Day 1 - Quadratic Graphs Properties REVIEW 1. Domain: 2. Range: Function: values never repeat When given a graph use the line test to see if it is a function. 4. Sketch and label y = x and y = - x above. Describe the difference between the graphs. Quadratic Function is a function that can be written in the form y = ax + bx + c, a 0. y = 2x y = 3x y = x + 2x - 3 *** It has a squared term *** Simplest quadratic form(parent graph): y = x 1. Use the table of values to graph y = x x y = x 2-1 Check your graph on the calculator! 1 = x The graph of a quadratic function is called a and is shaped like a. The POINT where the graph changes direction is called the or.

3 Recall what happens to the absolute value parent graph when you negate the coefficient of x? The graph over the x-axis. What do you think will happen to the quadratic parent graph y = x when we negate the squared term? 2. Use the table of values to graph y = -x 2 x y -1 2 is positive (+) the graph opens + + we have a parabola. The vertex is the point. 2 is negative (-) the graph opens - - we have a parabola. The vertex is the point.

4 Label the vertex and determine if it is a maximum (highest) minimum (lowest) point

5 Graphing a Quadratic Function using a Table of Values and/or use your calculator ("y =") 3. Use the table of values to graph y = 2x x y -1 What is the vertex (turning point)? MAX/MIN is greater than 1 the graph is than the original graph y = x 4. Use the table of values to graph y = ½x x y -1 What is the vertex (turning point)? MAX/MIN is less than 1 the graph is than the original graph y = x

6 Graphing a Quadratic Function using a Table of Values and/or use your calculator ("y =") 5. Use the table of values to graph y = x + 3 x y -1 What is the vertex (turning point)? MAX/MIN When we add a constant "c" to the quadratic term x the original graph y = x is shifted by "c" units (y-intercept) 6. Use the table of values to graph y = x - 3 x y -1 What is the vertex (turning point)? MAX/MIN When we subtract a constant "c" from the quadratic term x the original graph y = x is shifted by "c" units (y-intercept)

7 Quadratic Functions and Equations Day 1 - Homework Fill in the blank with the appropriate word, phrase or symbol to make a true statement. 1. A function is any function that can be written in the form y = ax + bx + c. 2. The graph of a quadratic function is a u-shaped curve called a. 3. The point where the parabola changes direction is called the. 4. If a parabola opens upward, then the coefficient of the squared term is and the vertex is the point. 5. If a parabola opens downward, then the coefficient of the squared term is and the vertex is the point. Without graphing, describe how each graph differs from the parent graph y = x 8. 9.

8 Sketch your graphs based on the shift rules.

9 Quadratic Functions and Equations Adding a constant to a quadratic term shifts the graph or. ketch the graph of the function indicated, draw in and label the vertex. y = x y = x + 2x y = x + 4x y = x + 6x Property of a Parabola (quadratic function) AOS The line where we fold the parabola is called the axis of symmetry (AOS). called. The axis of symmetry is a line which always passes through the graph at its. REMEMBER: x = # AOS

10 Referring to the last page, describe the relationship between the turning point (vertex) and the axis of symmetry for each graph? Now that you realize the relationship, you can write the equation of the axis of symmetry without a graph... if you know the vertex. 2. If the vertex of a parabola is (-3, 2), what is the equation of the axis of symmetry? 3. Given a quadratic function whose axis of symmetry is x = - 5, name a point that could possibly be the vertex of the parabola. We can even write the equation of the axis of symmetry without using a graph... if you know the equation of the quadratic function. In a quadratic function y = ax + bx + c The equation for the axis of symmetry is x = -b Sample Question: a) Graph the function y = x 2-2x + 1, using a table of values. b) What is the equation of the axis of symmetry? y-value?

11 Calculate the vertex without using the calculator or the graph? 1. From the axis of symmetry we know the vertex's x-value 3. Write as a point (x, y) to label the the vertex. Example 1: Given the function y = x 2-6x + 4, what is the equation of the axis of symmetry? Therefore, what is the x-coordinate of the vertex? Substitute this value into the function y = x 2-6x + 4 and solve for y. What is the vertex (turning point)? Example 2: Given the function y = 2x 2 + 4x - 1, what is the equation of the axis of symmetry? Therefore, what is the x-coordinate of the vertex? Substitute this value into the function y = 2x 2 + 4x - 1 What is the vertex (turning point)? Example 3: Given the function y = x - 8x - 7 What is the equation of the axis of symmetry? What is the vertex (turning point)? Remember: f(x) = y f(x) So we could write the function in example 3 as f(x) = x - 8x - 7 Example 4: "+" for =.5 Given the function f(x) = 3x - 9x + 2 What is the equation of the axis of symmetry? What is the vertex (turning point)? Example 5: Given the function f(x) = -x 2-6x - 10 What is the equation of the axis of symmetry? What is the vertex (turning point)?

12 Quadratic Functions and Equations Day 2 - Homework Example 1: Given the function + 8x + 9 What is the equation of the axis of symmetry? What is the vertex (turning point)? Example 2: Given the function f(x) = -x + 4x + 3 What is the equation of the axis of symmetry? What is the vertex (turning point)? Example 3: Given the function y = 2x - 20x What is the equation of the axis of symmetry? What is the vertex (turning point)? Example 4: Given the function y = 3x What is the equation of the axis of symmetry? What is the vertex (turning point)? Example 5: Given the function y = x - 3x -10 What is the equation of the axis of symmetry? What is the vertex (turning point)? Problem 6 and 7: Label the axis of symmetry and vertex for each graph

13 Quadratic Functions and Equations Day 3 - Finding Roots Graphically Solve the following systems of equations algebraically using substitution. 1. y = x y = -x + 3 y = 3x - 6 y = 0 REMEMBER: The solution for a system of equations is the of of the two graphs. EXAMPLE: y = x - 5x + 6 What type of function is this? What type of graph? What type of function is this? What type of graph? How do you think we would begin to solve the following system of equations algebraically? Again, what is the solution to a system of equations? The of. Given the following graph of the quadratic-linear system, how many solutions are there? Circle and label the point of intersection on the graph. y = x - 5x + 6 y = 0

14 Quadratic Function is a function that can be written in the form + bx + c -or- f(x) = ax + bx + c, where a 0. Quadratic Equation can be written as ax 2 + bx + c = 0, where a 0 The solutions of a quadratic equation are the x-intercepts, (the values of x where the graph crosses(intersects) the x-axis(y=0)) They are also known as the roots or zeros of the function **There can be zero, one or two solutions for a quadratic equation** Circle and label the point(s) of intersection on each graph, then identify the root(s) (x values) of the quadratic equation. 1. solution(s): 2. root(s): 3. zero(s): R TS 4. Solution(s): 5. Root(s): 6. Zero(s): 7. Root(s):

15 When using your calculator to find the solution(s) to a quadratic equation we want to look at the graph or the table associated with the graph... but how do we enter it in the "y =" screen when there is no "y"? EXAMPLE: x - 9x = 0 Remember that this is the combination of two equations y = x - 9x 0 1. Enter these two equations as your y and y 2. GRAPH and find the point(s) of intersection using 2nd TRACE 5:Intersect. The solutions of the quadratic equation are: Use your calculator to solve the following quadratic equations x - 4 = 0 Solutions: 2. - x - 6 = 0 Roots: 3. x - 7x + 10 = 0 Zeros: 4. x + 5x + 4 = 0 Solutions: 5. -x + 6x = 0 Roots: 6. -x - 2x + 5 = 0 Zeros: x - 14 = 0 Solutions: 1. y = x 2 + 7x + 6 Number of Solutions 2. y = x 2-6x + 9 Number of Solutions Solution(s) Solution(s) Factor: 2 + 7x + 6 Factor: 2-6x y = -x 2-3 Number of Solutions Solution(s) Factor: - 2-3

16 Quadratic Functions and Equations Day 3 - Homework Use your calculator to solve the following quadratic equations. 1. x - 4x + 3 = 0 Solutions: 2. x - 4x - 21 = 0 Roots: 3. 2x + 5x - 3 = 0 Zeros: 4. x - 64 = 0 Solutions: 5. x - 2x = 0 Roots: y = x - 4x + 4 Number of Solutions Solution(s) Factor: x - 4x + 4 y = x - x - 6 Number of Solutions Solution(s) Factor: x - x - 6 y = x + 2 Number of Solutions Solution(s) Factor: x + 2 Circle and label the points of intersection and then identify the roots of the quadratic equation. 9. solutions: 10. roots:

17 Quadratic Functions and Equations Day 4 - Solving Quadratic Equations for Zero (Algebraically) 1. Factor Completely: 3x - 6x - 24x 2. Solve for x: 5(x - 2) = 0 Principle of Zero Property: if a x b = 0 then either a = 0 or b = 0 or both a and b must be zero. therefore, if (x - 5)(x + 2) = 0 (x - 5) = 0 or (x + 2) = 0 or both must be zero x - 5 = 0 x + 2 = x = 5 or x = - 2 Quadratic Equation can be written as ax 2 + bx + c = 0, where a 0 The solutions of a quadratic equation are the x-intercepts, (the values of x where the graph crosses(intersects) the x-axis(y=0)) They are also known as the roots or zeros of the function **There can be zero, one or two solutions for a quadratic equation**

18 Example 1: a) What are the solutions for the graph? b) Given the system of equations y = x + 8x + 7 = 0 by substitution method, we get the quadratic function x + 8x + 7 = 0 x + 8x + 7 = 0 (x + 1)(x + 7) = 0 d) Set each factor equal to zero x + 1 = 0 or x + 7 = 0 e) Solve each equation for x x = -1 x = - 7 f) Solution Set { -1, -7 } EXAMPLES:

19 Practice Problems Solve for the zeros of the equations algebraically. (check on your calculator) (x 3)(x + 7) = 0 9. x 2-4x + 3 = 0 (x + 5)(x + 8) = x 2 + 6x 16 = 0 (x 7)(x 1) = x 2 1 = 0 (x + 2)(x 4) = x - 7 x 15 = 0 5x(x 2) = x 2-63 = 0 6x 15 = x 3-20x 2 60x = 0 ** (there are 3 roots!) 9x 2 25 = x 2-22x +48 = x 20 = x 2 +7x 5 = 0

20 Quadratic Functions and Equations Day 4 - Homework Solving Quadratic Equations by Factoring 1. 2x (x + 5) = 0 7. x - 5x = 0 2. (n - 4)(n + 4) = 0 8. x +5x + 4 = 0 3. (k - 3)(k + 5) = x - 27 = 0 4. (2x - 1)(x + 9) = x + 30x = 0 5. x - 9x +18 = 0 3k - 18k - 21 = 0 6. m - 64 = x - 42x + 72 = 0

21 Quadratic Functions and Equations Day 5 - Solving Quadratic Equations : x 2-2y = 4 y = 0 b) What is the equation of the axis of symmetry? c) What is the vertex? d) What are the roots (x-intercepts/solutions)? 1. x + x - 30 = 0 2. x + 12x + 27 = = n - 99 = x - 16 = y = 0 What do all of these problems have in common?

22 Review: When we need to enter a quadratic function on the calculator we solve for y first, so it is in the form y = ax 2 + bx + c. When we are solving for the roots/solutions of a quadratic equation, we know y = 0 because we are looking for the x- in the form 0 = ax 2 + bx + c. How would we solve a quadratic equation algebraically x + 2x = 3? x + 2x - 3 = 0 (x + 3)(x - 1) = 0 3. Set each factor equal to zero and solve for x. x + 3 = 0 x - 1 = x = -3 x = 1 {-3, 1} x 2 2. x 2 = 2x p + 21 = n - 10 = = - x 2 4. k + 15k = r = r 2

23 Quadratic Functions and Equations Day 5 - Homework 1. x 2-9 = y 2-10 = x x = = 4a 3. x 2 + 6x - 4 = x 2-9 = x x + 49 = 3x 9. 6x - 9 = - 3x 5. 2 = n 10. 3x 2-10 = 13x

24 Quadratic Functions and Equations Day 6 - Solving Quadratic Equations Graphically - Calculator : 1. Graph the following system of linear equations and find the solution. y = 2x - 1 2y + x = x + 2x = 0 2. x + 5x - 35 = 3x Solve then graph each quadratic equation. Circle and label the roots on each graph. 3. x + 2x = 8 4. x = 48

25 (Short Cut) Graph the system of linear/quadratic equations. Circle and label the solutions (POI). 1. y = x - x y = x - x y = 0 y = 2 Solution(s) Solution(s) Solve each quadratic equation x - 2 = 0 4. x - x = 2

26 1. x 2. k + 15k = m = - 16m x = x 5. 2k - 14 = -3k 6. m = 3m 7. 3v + 36v + 49 = 8v 8. k + 15k = x - 26x = k + 80 = - 80p

27 Quadratic Functions and Equations Day 6 - Homework

28 Quadratic Functions and Equations 1. What is the Domain? What is the Range? 2. What is the Domain? What is the Range? 1.

29 2. 3.

30 Let's graph it! x y Quadratic Function:

31 Identifying Function Graphs Day 7 - Homework THINK: Identify each type of graph and explain which one you would choose to represents how you would like to be paid over the next year? a) b) c) d) a) b) c) d)

32 a) y = x + 5 b) y = -2 x + 3 c) y = 3 x d) 2x = y - 4 e) y = (1/2) x f) 5 = 4x - x

33 Quadratic Functions and Equations Review Day 8 - Classwork & Homework c) is the vertex a MAX / MIN?

34 quadratic function: y = x + 4x - 12 b) What are the roots of the equation x + 4x - 12 =0

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36 Quadratic Function:

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45 Explore the Quadratic Equation.webarchive

46 Quadratic Functions and Equations Day 6 - Homework THINK: Identify each type of graph and explain which one you would choose to represents how you would like to be paid over the next year? a) b) c) d) a) b) c) d)

47 a) y = x + 5 b) y = -2 x + 3 c) y = 3 x d) 2x = y - 4 e) y = (1/2) x f) 5 = 4x - x

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